Abstract Probabilistic sentential decision diagrams are logic circuits where the inputs of disjunctive gates are annotated by probability values. They allow for a compact representation of joint probability mass functions defined over sets of Boolean variables, that are also consistent with the logical constraints defined by the circuit. The probabilities in such a model are usually “learned” from a set of observations. This leads to overconfident and prior-dependent inferences when data are scarce, unreliable or conflicting. In this work, we develop the credal sentential decision diagrams, a generalisation of their probabilistic counterpart that allows for replacing the local probabilities with (so-called credal) sets of mass functions. These models induce a joint credal set over the set of Boolean variables, that sharply assigns probability zero to states inconsistent with the logical constraints. Three inference algorithms are derived for these models. These allow to compute: (i) the lower and upper probabilities of an observation for an arbitrary number of variables; (ii) the lower and upper conditional probabilities for the state of a single variable given an observation; (iii) whether or not all the probabilistic sentential decision diagrams compatible with the credal specification have the same most probable explanation of a given set of variables given an observation of the other variables. These inferences are tractable, as all the three algorithms, based on bottom-up traversal with local linear programming tasks on the disjunctive gates, can be solved in polynomial time with respect to the circuit size. The first algorithm is always exact, while the remaining two might induce a conservative (outer) approximation in the case of multiply connected circuits. A semantics for this approximation together with an auxiliary algorithm able to decide whether or not the result is exact is also provided together with a brute-force characterization of the exact inference in these cases. For a first empirical validation, we consider a simple application based on noisy seven-segment display images. The credal models are observed to properly distinguish between easy and hard-to-detect instances and outperform other generative models not able to cope with logical constraints.

中文翻译：

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凭据句子决策图中的可追踪推理

摘要 概率句子决策图是逻辑电路，其中析取门的输入由概率值注释。它们允许在布尔变量集上定义的联合概率质量函数的紧凑表示，这也与电路定义的逻辑约束一致。这种模型中的概率通常是从一组观察中“学习”出来的。当数据稀缺、不可靠或相互冲突时，这会导致过度自信和先验依赖的推断。在这项工作中，我们开发了 credal 句子决策图，这是它们概率对应物的概括，允许用（所谓的 credal）质量函数集替换局部概率。这些模型在布尔变量集上引入了联合信任集，这将零概率分配给与逻辑约束不一致的状态。为这些模型导出了三种推理算法。这些允许计算：（i）任意数量变量的观察的下限和上限概率；(ii) 给定观察的单个变量状态的下限和上限条件概率；(iii) 是否所有与信用规范兼容的概率句子决策图都对给定的一组变量具有相同的最可能的解释给定对其他变量的观察。这些推论是易于处理的，因为所有三种算法都基于自下而上的遍历，并在分离门上执行局部线性规划任务，可以在多项式时间内解决电路大小问题。第一个算法总是精确的，而在多连接电路的情况下，其余两个可能会引起保守（外部）近似。在这些情况下，还提供了这种近似的语义以及能够确定结果是否准确的辅助算法以及精确推理的蛮力表征。对于第一个经验验证，我们考虑一个基于嘈杂七段显示图像的简单应用程序。观察到信用模型可以正确区分容易检测和难以检测的实例，并且优于其他无法处理逻辑约束的生成模型。在这些情况下，还提供了这种近似的语义以及能够确定结果是否准确的辅助算法以及精确推理的蛮力表征。对于第一个经验验证，我们考虑一个基于嘈杂七段显示图像的简单应用程序。观察到信用模型可以正确区分容易检测和难以检测的实例，并且优于其他无法处理逻辑约束的生成模型。在这些情况下，还提供了这种近似的语义以及能够决定结果是否准确的辅助算法以及精确推理的蛮力表征。对于第一个经验验证，我们考虑一个基于嘈杂七段显示图像的简单应用程序。观察到信用模型可以正确区分容易检测和难以检测的实例，并且优于其他无法处理逻辑约束的生成模型。